I'm trying to learn about inexact differentials, and got stuck on step (147) here: http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html where it claims that
$Y'\frac{\partial\sigma}{\partial x} = X' \frac{\partial\sigma}{\partial y} \color{red}{= \frac{X'Y'}{\tau}}$
where $\sigma$ is a curve s.t. $\frac{\mathrm dy}{\mathrm dx}$ is constant along it and $\tau$ is a function.
My question is how the last equality can be reached; is it given that there exists a function $\tau = \frac{X'}{\frac{\partial\sigma}{\partial x}} = \frac{Y'}{\frac{\partial\sigma}{\partial y}}$ under these conditions? I don't see any particular reason why $\frac{X'}{\frac{\partial\sigma}{\partial x}} \overset{\color{red}{?}}{=} \frac{Y'}{\frac{\partial\sigma}{\partial y}}$ should hold.